Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13) x 2 − 5000.002 x + 10 Compute percent relative errors for your results.

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Answer 1

Textbook Question

Chapter 3, Problem 11P

Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13)

$x 2 − 5000.002 x + 10$

Compute percent relative errors for your results.

Expert Solution & Answer

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

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Answer 2

Textbook Question

Chapter 3, Problem 11P

Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13)

$x 2 − 5000.002 x + 10$

Compute percent relative errors for your results.

Expert Solution & Answer

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

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Answer 3

Textbook Question

Chapter 3, Problem 11P

Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13)

$x 2 − 5000.002 x + 10$

Compute percent relative errors for your results.

Expert Solution & Answer

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

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Answer 4

Textbook Question

Chapter 3, Problem 11P

Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13)

$x 2 − 5000.002 x + 10$

Compute percent relative errors for your results.

Expert Solution & Answer

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

Answer 8

Expert Solution & Answer

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

To calculate: The roots of the equation $x2−5000.002x+10$ with 5-digit arithmeticchopping by the use of the formula $x=−b±b2−4ac2a$ and $x=−2cb±b2−4ac$ . Also, find the percent relative error.

Answer 12

Answer to Problem 11P

Solution:

The roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−b±b2−4ac2a$ are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

And, the roots of the equation $x2−5000.002x+10$ by the use of the formula $x=−2cb±b2−4ac$ are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of approximately $0%$ .

Answer 13

Explanation of Solution

Given:

The equation, $x2−5000.002x+10$ .

Formula used:

The quadratic formula for the equation $ax2+bx+c=0$ is,

$x=−b±b2−4ac2a$

Alternative formula for roots when $b2>>4ac$ ,

$x=−2cb±b2−4ac$

Relative error formula:

$Relative error = | true value−approximate valuetrue value |$

Calculation:

Consider the following equation,

$x2−5000.002x+10=0$

Here, $a=1,b=−5000.002 and c=10$ .

Therefore, the roots of the equation from the quadratic formula is,

$x=5000.002±(5000.002)2−4×102=5000.002±(5000.002)2−4×102=5000.002±4999.9982$

Thus,

$x1=5000.002+4999.9982=100002=5000$

And,

$x2=5000.002−4999.9982=0.0042=0.002$

Now, chop to 5 digits and find the root as below,

$x=5000.0±(5000.0)2−4×102=5000.0±25000000−4×102=5000.0±249999602=5000.0±4999.92$

Solve for two different roots:

$x1′=5000.0+4999.92=9999.92=4999.9$

And,

$x2′=5000.0−4999.92=0.12=0.05$

Hence, the relative percent error of the first root $4999.9$ is,

$Error=|5000−4999.95000|×100%=|0.15000|×100%=0.002%$

And, the relative percent error of the second root $0.05$ is,

$Error=|0.002−0.050.002|×100%=|−0.0480.002|×100%=2400%$

Hence, the roots of the equations are $4999.9$ with relative percent error of $0.002%$ and $0.05$ with relative percent error of $2400%$ .

Now, for the provided equation $b2>>4ac$ , Therefore, use alternative formula to find the roots by chopping to 5-digit,

$−2cb±b2−4ac=−2(10)−5000.0±(5000.0)2−4×10=−2(10)−5000.0±25000000−4×10=−20−5000.0±24999960=−20−5000.0±4999.9$

Solve for two different roots,

$x1′′=−20−5000.+4999.9=−20−0.1=200$

And,

$x1′′=−20−5000.0−4999.9=−20−9999.9=0.002$

Therefore, the relative percent error of the first root $200$ is,

$Error=|5000−2005000|×100%=|48005000|×100%=96%$

And, the relative percent error of the second root $0.00200002$ is,

$Error=|0.002−0.0020.002|×100%=|00.002|×100%=0%$

Hence, the roots of the equations are $200$ with relative percent error of $96%$ and $0.002$ with relative percent error of $0%$ .

Answer 14

Ch. 3 - Convert the following base-2 numbers to base-10:...Ch. 3 - Convert the following base-8 numbers to base-10:...Ch. 3 - 3.3 Compose your own program based on Fig. 3.11...Ch. 3 - In a fashion similar to that in Fig. 3.11, write a...Ch. 3 - 3.5 The infinite series converges on a value...Ch. 3 - 3.6 Evaluate using two approaches and and...Ch. 3 - The derivative of f(x)=1/(13x2) is given by...Ch. 3 - 3.8 (a) Evaluate the polynomial at . Use...Ch. 3 - Calculate the random access memory (RAM) in...Ch. 3 - Determine the number of terms necessary to...

Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13) x 2 − 5000.002 x + 10 Compute percent relative errors for your results.

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